6.2: Phenomological Graph
- Page ID
- 25450
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The excerpt of data you have been given includes the world’s population in billions, by year. That is all. Figure \(\PageIndex{1}\) shows the data plotted in a phenomenological way— population size versus year, supplemented with a curve going back 2000 years to provide perspective. The blue dots show the range of data you will be using to project the future population, and the black ‘×’ marks a great demographic transition that is not obvious in this graph, but that will become glaringly so in Figure 6.3.1.
Can you project global population by simply extending that curve? The population is clearly rising at an enormous rate, expanding most recently from 3 billion to 7 billion in less than half a century. Simply projecting the curve would lead to a prediction of more than 11 billion people by the middle of the 21th century, and more than 15 billion by the century’s end.
But such an approach is too simplistic. In one sense, the data are all contained in that curve, but are obscured by the phenomena themselves. We need to extract the biology inherent in the changing growth rate r as well as the ecology inherent in the changing density dependence s. In other words, we want to look at data showing 1/N ∆N /∆t versus N, as in Figure 4.4.1.
Table 6.1.1 shows a subset of the original data, t and N, plus calculated values for ∆N, ∆t, and 1/N ∆N /∆t. In row 1, for example, ∆N shows the change in N between row 1 and row 2: 0.795−0.606 = 0.189 billion. Likewise, ∆t in row 1 shows how many years elapse before the time of row 2: 1750 − 1687 = 63 years. The final column in row 1 shows the value of 1/N ∆N /∆t : 1/0.606 × 0.189/63 = 0.004950495..., which rounds to 0.0050. Row 21 has no deltas because it is the last row in the table.
| Point | Year t | N billions | ∆N | ∆t | \(\frac{1}{N}\frac{∆N}{∆t}\) |
|---|---|---|---|---|---|
| 1. | 1687 | 0.606 | 0.189 | 63 | 0.0050 |
| 2. | 1750 | 0.795 | 0.174 | 50 | 0.0044 |
| 3. | 1800 | 0.969 | 0.296 | 50 | 0.0061 |
| 4. | 1850 | 1.265 | 0.391 | 50 | 0.0062 |
| 5. | 1900 | 1.656 | 0.204 | 20 | 0.0062 |
| 6. | 1920 | 1.860 | 0.210 | 10 | 0.0113 |
| 7. | 1930 | 2.070 | 0.230 | 10 | 0.0111 |
| 8. | 1940 | 2.300 | 0.258 | 10 | 0.0112 |
| 9. | 1950 | 2.558 | 0.224 | 5 | 0.0175 |
| 10. | 1955 | 2.782 | 0.261 | 5 | 0.0188 |
| 11. | 1960 | 3.043 | 0.307 | 5 | 0.0202 |
| 12. | 1965 | 3.350 | 0.362 | 5 | 0.0216 |
| 13. | 1970 | 3.712 | 0.377 | 5 | 0.0203 |
| 14. | 1975 | 4.089 | 0.362 | 5 | 0.0177 |
| 15. | 1980 | 4.451 | 0.405 | 5 | 0.0182 |
| 16. | 1985 | 4.856 | 0.432 | 5 | 0.0178 |
| 17. | 1990 | 5.288 | 0.412 | 5 | 0.0156 |
| 18. | 1995 | 5.700 | 0.390 | 5 | 0.0137 |
| 19. | 2000 | 6.090 | 0.384 | 5 | 0.0126 |
| 20. | 2005 | 6.474 | 0.392 | 5 | 0.0121 |
| 21. | 2010 | 6.866 |